We construct virtual fundamental classes on nested Hilbert schemes of points and curves in complex nonsingular projective surfaces. These classes recover the virtual classes of Seiberg-Witten theory as well as the (reduced) stable theory, and play a crucial role in the reduced Donaldson-Thomas theory of local-surface-threefolds that we study in [GSY17b] (arXiv:1807.05697). We show that certain integrals against the virtual fundamental classes of punctual nested Hilbert schemes are expressed as integrals over the products of the Hilbert scheme of points. We are able to find explicit formulas for some of these integrals by relating them to Carlsson-Okounkov's vertex operator formulas.

中文翻译：

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曲面上的嵌套希尔伯特方案：虚拟基础类

我们在复杂的非奇异射影面上的点和曲线的嵌套希尔伯特方案上构造虚拟基本类。这些类恢复了 Seiberg-Witten 理论的虚拟类以及（简化的）稳定理论，并且在我们在 [GSY17b] (arXiv:1807.05697) 中研究的局部表面三重的简化 Donaldson-Thomas 理论中起着至关重要的作用）。我们表明，针对准点嵌套希尔伯特方案的虚拟基本类的某些积分表示为希尔伯特点方案的乘积上的积分。通过将这些积分与 Carlsson-Okounkov 的顶点算子公式相关联，我们能够找到其中一些积分的明确公式。